A novel way of solving the dictionary learning problem is proposed in this paper. It is based on a so-called direct optimization as it avoids the usual technique which consists in alternatively optimizing the coefﬁcients of a sparse decomposition and in optimizing dictionary atoms. The algorithm we advocate simply performs a joint proximal gradient descent step over the dictionary atoms and the coefﬁcient matrix. After having derived the algorithm, we also provided in-depth discussions on how the stepsizes of the proximal gradient descent have been chosen. In addition, we uncover the connection between our direct approach and the alternating optimization method for dictionary learning. We have shown that it can be applied to a broader class of non-convex optimization problems than the dictionary learning one. As such, we have denoted the algorithm as a one-step blockcoordinate proximal gradient descent. The main advantage of our novel algorithm is that, as suggested by our simulation study, it is more efﬁcient than alternating optimization algorithms.
Direct Optimization of the dictionary learning problem